Preference Restrictions in Computational Social Choice: A Survey

TL;DR

The survey systematically catalogs restricted preference domains—notably single-peaked, single-crossing, Euclidean, and tree/circle variants—and analyzes how these constraints affect majority structure, strategyproofness, recognition, and winner determination. It synthesizes foundational theory with algorithmic results, detailing linear-time recognizers, sampling methods, and complexity classifications, while also presenting new findings on nearly-structured preferences and forbidden-subprofile characterizations. A unifying theme is identifying islands of tractability where classic NP-hard winner-determination problems become polynomial, or where recognition certificates enable efficient use of specialized algorithms. The work also surveys gaps and open questions, highlighting the interplay between economic insights and computational techniques in understanding structured voting data. Overall, it provides a comprehensive toolkit for researchers and practitioners to analyze, recognize, and compute outcomes under restricted preference models across single-winner and multi-winner settings.

Abstract

Social choice becomes easier on restricted preference domains such as single-peaked, single-crossing, and Euclidean preferences. Many impossibility theorems disappear, the structure makes it easier to reason about preferences, and computational problems can be solved more efficiently. In this survey, we give a thorough overview of many classic and modern restricted preference domains and explore their properties and applications. We do this from the viewpoint of computational social choice, letting computational problems drive our interest, but we include a comprehensive discussion of the economics and social choice literatures as well. Particular focus areas of our survey include algorithms for recognizing whether preferences belong to a particular preference domain, and algorithms for winner determination of voting rules that are hard to compute if preferences are unrestricted.

Figures (21)

• Figure 1: Some domain restrictions that guarantee a weak Condorcet winner, and inclusion relationships.
• Figure 2: Votes $v_1$ and $v_2$, shown as solid lines, are single-peaked with respect to $\lhd$. The vote $v_3$, depicted as a dashed line, is not single-peaked with respect to $\lhd$. While a profile consisting of $v_2$ and $v_3$ is single-peaked ($d$ and $e$ have to be flipped on $\lhd$), there is no ordering of the candidates for which the profile $(v_1,v_3)$ is single-peaked.
• Figure 3: A valley.
• Figure 4: A 2-Euclidean embedding of the Condorcet profile.
• Figure 5: Boundaries of non-$d$-Euclidean profiles with a given number of voters and alternatives. Each colored bullet point denotes the existence of such a non-Euclidean profile for the corresponding dimension. Figure reproduced from bulteau20222dimensional.

Theorems & Definitions (148)

• Definition 1
• Definition 2
• Definition 3
• Proposition 3.1
• proof
• Theorem 3.2: CaKe02CaKe16
• Definition 4
• Proposition 3.3
• proof
• Proposition 3.4: Median Voter Theorem